Question: The equation of hyperbola $H$ is $\dfrac {(y-8)^{2}}{49}-\dfrac {(x-9)^{2}}{16} = 1$. What are the asymptotes?
We want to rewrite the equation in terms of $y$ , so start off by moving the $y$ terms to one side: $\dfrac {(y-8)^{2}}{49} = 1 + \dfrac {(x-9)^{2}}{16}$ Multiply both sides of the equation by $49$ $(y-8)^{2} = { 49 + \dfrac{ (x-9)^{2} \cdot 49 }{16}}$ Take the square root of both sides. $\sqrt{(y-8)^{2}} = \pm \sqrt { 49 + \dfrac{ (x-9)^{2} \cdot 49 }{16}}$ $ y - 8 = \pm \sqrt { 49 + \dfrac{ (x-9)^{2} \cdot 49 }{16}}$ As $x$ approaches positive or negative infinity, the constant term in the square root matters less and less, so we can just ignore it. $y - 8 \approx \pm \sqrt {\dfrac{ (x-9)^{2} \cdot 49 }{16}}$ $y - 8 \approx \pm \left(\dfrac{7 \cdot (x - 9)}{4}\right)$ Add $8$ to both sides and rewrite as an equality in terms of $y$ to get the equation of the asymptotes: $y = \pm \dfrac{7}{4}(x - 9)+ 8$